theorem Th21:
  |{p1,p5,p9}| = 0 implies
  |{p1,p5,p7}| * |{p2,p5,p9}| = - |{p1,p2,p5}| * |{p5,p9,p7}|
  proof
    assume
A1: |{p1,p5,p9}| = 0;
A2: |{p1,p2,p5}| = |{p5,p1,p2}| & |{p1,p5,p9}| = - |{p5,p1,p9}| &
    |{p5,p1,p7}| = - |{p1,p5,p7}| & |{p5,p2,p9}| = - |{p2,p5,p9}| &
    |{p5,p9,p7}| = - |{p5,p7,p9}| by ANPROJ_8:29,ANPROJ_8:30,Th01; then
A3: |{p5,p1,p9}| = 0 by A1;
    |{p5,p1,p7}| * |{p5,p2,p9}| - |{p5,p1,p2}| * |{p5,p7,p9}| +
      |{p5,p1,p9}| * |{p5,p7,p2}| = 0 by ANPROJ_8:28;
    hence thesis by A2,A3;
  end;
